#8695334

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#37214	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,4,5)(3,7,8,6)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,22)(21,23)(25,26,31,28)(27,32,29,33)(30,34,38,36)(35,39,37,40);(2,5)(6,7)(9,11,15,13)(10,16,12,18)(14,19,22,21)(17,23,20,24)(25,27,31,29)(26,32,28,33)(30,35,38,37)(34,39,36,40);(1,2)(3,6)(4,5)(7,8)(25,27,31,29)(26,33,28,32)(30,35,38,37)(34,40,36,39);(1,2)(3,6)(4,5)(7,8)(9,12,15,10)(11,16,13,18)(14,17,22,20)(19,24,21,23)(25,28,31,26)(27,32,29,33)(30,36,38,34)(35,39,37,40);(1,3)(2,6)(4,8)(5,7)(9,13,15,11)(10,16,12,18)(14,19,22,21)(17,24,20,23)(25,29,31,27)(26,32,28,33)(30,37,38,35)(34,39,36,40);(9,14,15,22)(10,17,12,20)(11,19,13,21)(16,23,18,24)(25,30,31,38)(26,34,28,36)(27,35,29,37)(32,39,33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=hi,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8602530);